On the Convergence of a Modified Kaehler-Ricci flow

We study the convergence of a modified Kaeher-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang

On local holomorphic maps preserving invariant (p,p)-forms between bounded symmetric domains

Let $D, \Omega_1, ..., \Omega_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $\Omega_1 \times... \Omega_m$ preserving the invariant $(p, p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank one case for any $p$ and in the rank at least two case for certain sufficiently large $p$. The total geodesy thus follows if $D=\mathbb{B}^n, \Omega_i = \mathbb{B}^{N_i}$ for any $p$ or if $D=\Omega_1 =...=\Omega_m$ with rank$(D)\geq 2$ and $p$ sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties $D / \Gamma$ preserving invariant $(p, p)$-forms is modular, where $\Gamma$ is a torsion free, discrete, finite co-volume subgroup of Aut$(D)$. This solves partially a problem raised by Mok

Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass

This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case [Auchmuty and Beals, Arch. Ration. Mech. Anal., 1971] to the non-isentropic case. Compared with the previous result of variational rotating solutions in non-isentropic case [Wu, Journal of Differential Equations, 2015], to keep the constraint of a prescribed finite total mass, the author establishes a new variational structure the non-isentropic Euler-Poisson equations

Asymptotic stability of shock waves and rarefaction waves under periodic perturbations for 1-D convex scalar conservation laws

In this paper we study large time behaviors toward shock waves and rarefaction waves under periodic perturbations for 1-D convex scalar conservation laws. The asymptotic stabilities and decay rates of shock waves and rarefaction waves under periodic perturbations are proved

Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-d convex scalar viscous conservation laws

This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $L^\infty(\mathbb{R})$ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shift. And the vanishing viscosity limit for the shift is also shown. The key elements of the poof consist of the construction of an ansatz which tends to two periodic solutions as $x \rightarrow \pm\infty,$ respectively, and the anti-derivative variable argument, and an elaborate use of the maximum principle. For the rarefaction wave, we also show the stability in the $L^\infty(\mathbb{R})$ norm.Comment: 43 pages, 3 figure

The CR immersion into a sphere with the degenerate CR Gauss map

It is a classical problem in algebraic geometry to characterize the algebraic subvariety by using the Gauss map. In this note, we try to develop the analogue theory in CR geometry. In particular, under some assumptions, we show that a CR map between spheres is totally geodesic if and only if the CR Gauss map of the image is degenerate

On the self-similar solution to full compressible Navier-Stokes equations without heat conductivity

In this work, we establish a class of globally defined, large solutions to the free boundary problem of compressible Navier-Stokes equations with constant shear viscosity and vanishing bulk viscosity. We establish such solutions with initial data perturbed arbound any self-similar solution when \gamma > 7/6. In the case when 7/6 < \gamma < 7/3, as long as the self-similar solution has bounded entropy, a solution with bounded entropy can be constructed. It should be pointed out that the solutions we obtain in this fashion do not in general keep being a small perturbation of the self-similar solution due to the second law of thermodynamics, i.e., the growth of entropy. If in addition, in the case when 11/9 < \gamma < 5/3, we can construct a solution as a global-in-time small perturbation of the self-similar solution and the entropy is uniformly bounded in time

Non-invasive dynamic or wide-field imaging through opaque layers and around corners

In turbid media, scattering of light scrambles information of the incident beam and represents an obstacle to optical imaging. Noninvasive imaging through opaque layers is challenging for dynamic and wide-field objects due to unreliable image reconstruction processes. We here propose a new perspective to solve these problems: rather than using the full point-spread-function (PSF), the wave distortions in scattering layers can be characterized with only the phase of the optical-transfer-function (OTF, the Fourier transform of PSF), with which diffraction-limit images can be analytically solved. We then develop a method that exploits the redundant information dynamic objects, and can reliably and rapidly recover OTFs' phases within several iterations. It enables not only noninvasive video imaging at 25 ~ 200 Hz of a moving object hidden inside turbid media, but also imaging under weak illumination that is inaccessible with previous methods. Furthermore, by scanning a localized illumination on the object plane, we propose a wide-field imaging approach, with which we demonstrate an application where a photoluminescent sample hidden behind four-layers of opaque polythene films is imaged with a modified multi-photon excitation microscopy setup.Comment: Revised version with a new experiment adde

ARJA: Automated Repair of Java Programs via Multi-Objective Genetic Programming

Recent empirical studies show that the performance of GenProg is not satisfactory, particularly for Java. In this paper, we propose ARJA, a new GP based repair approach for automated repair of Java programs. To be specific, we present a novel lower-granularity patch representation that properly decouples the search subspaces of likely-buggy locations, operation types and potential fix ingredients, enabling GP to explore the search space more effectively. Based on this new representation, we formulate automated program repair as a multi-objective search problem and use NSGA-II to look for simpler repairs. To reduce the computational effort and search space, we introduce a test filtering procedure that can speed up the fitness evaluation of GP and three types of rules that can be applied to avoid unnecessary manipulations of the code. Moreover, we also propose a type matching strategy that can create new potential fix ingredients by exploiting the syntactic patterns of the existing statements. We conduct a large-scale empirical evaluation of ARJA along with its variants on both seeded bugs and real-world bugs in comparison with several state-of-the-art repair approaches. Our results verify the effectiveness and efficiency of the search mechanisms employed in ARJA and also show its superiority over the other approaches. In particular, compared to jGenProg (an implementation of GenProg for Java), an ARJA version fully following the redundancy assumption can generate a test-suite adequate patch for more than twice the number of bugs (from 27 to 59), and a correct patch for nearly four times of the number (from 5 to 18), on 224 real-world bugs considered in Defects4J. Furthermore, ARJA is able to correctly fix several real multi-location bugs that are hard to be repaired by most of the existing repair approaches.Comment: 30 pages, 26 figure

Holomorphic isometry from a Kahler manifold into a product of complex projective manifolds

We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative